def test_branch_bug():
assert hyperexpand(hyper((-Rational(1, 3), Rational(1, 2)), (Rational(2, 3), Rational(3, 2)), -z)) == \
-z**Rational(1, 3)*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([Rational(7, 6), 1], [], [Rational(2, 3)], [Rational(1, 6), 0], z)) == \
2*z**Rational(2, 3)*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) -
2*lowergamma(Rational(2, 3), z)/z**Rational(2, 3))*gamma(Rational(2, 3))/gamma(Rational(5, 3))
def test_branch_bug():
assert hyperexpand(hyper((-Rational(1, 3), Rational(1, 2)), (Rational(2, 3), Rational(3, 2)), -z)) == \
-cbrt(z)*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([Rational(7, 6), 1], [], [Rational(2, 3)], [Rational(1, 6), 0], z)) == \
2*z**Rational(2, 3)*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) -
2*lowergamma(Rational(2, 3), z)/z**Rational(2, 3))*gamma(Rational(2, 3))/gamma(Rational(5, 3))
def test_branch_bug():
# TODO combsimp cannot prove that the factor is unity
assert powdenest(integrate(erf(x**3), x, meijerg=True).diff(x),
polar=True) == 2*erf(x**3)*gamma(Rational(2, 3))/3/gamma(Rational(5, 3))
assert integrate(erf(x**3), x, meijerg=True) == \
2*x*erf(x**3)*gamma(Rational(2, 3))/(3*gamma(Rational(5, 3))) \
- 2*gamma(Rational(2, 3))*lowergamma(Rational(2, 3), x**6)/(3*sqrt(pi)*gamma(Rational(5, 3)))
def test_branch_bug():
# TODO combsimp cannot prove that the factor is unity
assert powdenest(integrate(erf(x**3), x, meijerg=True).diff(x),
polar=True) == 2*erf(x**3)*gamma(Rational(2, 3))/3/gamma(Rational(5, 3))
assert integrate(erf(x**3), x, meijerg=True) == \
2*x*erf(x**3)*gamma(Rational(2, 3))/(3*gamma(Rational(5, 3))) \
- 2*gamma(Rational(2, 3))*lowergamma(Rational(2, 3), x**6)/(3*sqrt(pi)*gamma(Rational(5, 3)))
def test_specfun():
for f in [besselj, bessely, besseli, besselk]:
assert octave_code(f(n, x)) == f.__name__ + '(n, x)'
assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)'
assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)'
assert octave_code(airyai(x)) == 'airy(0, x)'
assert octave_code(airyaiprime(x)) == 'airy(1, x)'
assert octave_code(airybi(x)) == 'airy(2, x)'
assert octave_code(airybiprime(x)) == 'airy(3, x)'
assert octave_code(uppergamma(n, x)) == "gammainc(x, n, 'upper')"
assert octave_code(lowergamma(n, x)) == "gammainc(x, n, 'lower')"
assert octave_code(jn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2'
assert octave_code(yn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2'
assert octave_code(Chi(x)) == 'coshint(x)'
assert octave_code(Ci(x)) == 'cosint(x)'
assert octave_code(laguerre(n, x)) == 'laguerreL(n, x)'
assert octave_code(li(x)) == 'logint(x)'
assert octave_code(loggamma(x)) == 'gammaln(x)'
assert octave_code(polygamma(n, x)) == 'psi(n, x)'
assert octave_code(Shi(x)) == 'sinhint(x)'
assert octave_code(Si(x)) == 'sinint(x)'
assert octave_code(LambertW(x)) == 'lambertw(x)'
assert octave_code(LambertW(x, n)) == 'lambertw(n, x)'
assert octave_code(zeta(x)) == 'zeta(x)'
assert octave_code(zeta(
x, y)) == '% Not supported in Octave:\n% zeta\nzeta(x, y)'
def test_gamma():
k = Symbol('k', positive=True)
theta = Symbol('theta', positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, positive=True)
X = Gamma('x', k, theta)
assert simplify(E(X)) == k*theta
# can't get things to simplify on this one so we use subs
assert variance(X).subs({k: 5}) == (k*theta**2).subs({k: 5})
def test_gamma():
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, positive=True)
X = Gamma('x', k, theta)
assert simplify(E(X)) == k*theta
# can't get things to simplify on this one so we use subs
assert variance(X).subs({k: 5}) == (k*theta**2).subs({k: 5})
def test_uppergamma():
from diofant import meijerg, exp_polar, I, expint
assert uppergamma(4, 0) == 6
assert uppergamma(x, oo) == 0
assert uppergamma(x, y).diff(y) == -y**(x - 1) * exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
pytest.raises(ArgumentIndexError, lambda: uppergamma(x, y).fdiff(3))
assert uppergamma(S.Half, x) == sqrt(pi) * (1 - erf(sqrt(x)))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(Rational(1, 3), uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4 * pi * I) * x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \
gamma(y)*(1 - exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x) / x**2
assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x),
conjugate(y))
assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x))
assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo))
assert uppergamma(x, y).rewrite(expint) == y**x * expint(-x + 1, y)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z) * \
lowergamma(a - 1, z) - 1
# TODO [a+1, a+Rational(-1, 2)], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([Rational(1, 2), 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(5, 2)], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2))
assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([Rational(-1, 2) - 1, 1, 2], [Rational(1, 2), 3], z)) == \
sqrt(z)*(6*z/7 - Rational(6, 5))*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + Rational(1, 2), 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + Rational(1, 2))/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [Rational(1, 2), a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + Rational(1, 2))*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z) * \
lowergamma(a - 1, z) - 1
# TODO [a+1, a-S.Half], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([S.Half, 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(5, 2)], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2))
assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(6*z/7 - Rational(6, 5))*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + Rational(1, 2))/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + Rational(1, 2))*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_lowergamma():
assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert td(lowergamma(x, randcplx()), x)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
- meijerg([], [1, 1], [0, 0, x], [], y)
pytest.raises(ArgumentIndexError, lambda: lowergamma(x, y).fdiff(3))
assert lowergamma(Rational(1, 2), x) == sqrt(pi)*erf(sqrt(x))
assert not lowergamma(Rational(1, 2) - 3, x).has(lowergamma)
assert not lowergamma(Rational(1, 2) + 3, x).has(lowergamma)
assert lowergamma(Rational(1, 2), x, evaluate=False).has(lowergamma)
assert tn(lowergamma(Rational(1, 2) + 3, x, evaluate=False),
lowergamma(Rational(1, 2) + 3, x), x)
assert tn(lowergamma(Rational(1, 2) - 3, x, evaluate=False),
lowergamma(Rational(1, 2) - 3, x), x)
assert lowergamma(0, 1) == zoo
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(Rational(1, 3), lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))
assert lowergamma(
x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(
k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
assert (x*lowergamma(x, 1)/gamma(x + 1)).limit(x, 0) == 1
def test_sympyissue_4487():
assert simplify(integrate(exp(-x) * x**y, x)) == lowergamma(y + 1, x)
def test_sympyissue_15943():
s = Sum(binomial(n, k) * factorial(n - k), (k, 0, n))
assert s.doit().simplify() == E * (gamma(n + 1) - lowergamma(n + 1, 1))
def test_Mod1_behavior():
n = Symbol('n', integer=True)
# Note: this should not hang.
assert simplify(hyperexpand(meijerg([1], [], [n + 1], [0], z))) == \
lowergamma(n + 1, z)
def test_sympyissue_4487():
assert simplify(integrate(exp(-x)*x**y, x)) == lowergamma(y + 1, x)
def test_sympyissue_23156():
e = summation(1 / gamma(n), (n, 0, x))
assert e == E - E * x * lowergamma(x, 1) / gamma(x + 1)
assert e.limit(x, 0) == 0
assert e.subs({x: 0}) is nan
def test_meijerint():
s, t, mu = symbols('s t mu', extended_real=True)
assert integrate(
meijerg([], [], [0], [], s * t) *
meijerg([], [], [mu / 2], [-mu / 2], t**2 / 4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(
integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (erf(mu / (2 * sigma)) + 1)
assert c
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(
exp(-((x - mu) / sigma)**2 / 2) / sqrt(2 * pi * sigma**2), x, -oo,
oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0,
oo) == (Rational(1, 2), True)
# Test a bug
def res(n):
return (1 / (1 + x**2)).diff(x, n).subs({x: 1}) * (-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo),
meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
* meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, Rational(1, 2))/4).expand()
# Test hyperexpand bug.
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + Rational(1, 2),
alpha/2 + 1)), ((0, 0, Rational(1, 2)), (-Rational(1, 2),)), alpha**2/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - Rational(1, 2))*((-1)**s + 1)*gamma(s/2 + Rational(1, 2))/2
def test_Mod1_behavior():
from diofant import Symbol, simplify, lowergamma
n = Symbol('n', integer=True)
# Note: this should not hang.
assert simplify(hyperexpand(meijerg([1], [], [n + 1], [0], z))) == \
lowergamma(n + 1, z)
def test_meijerint():
s, t, mu = symbols('s t mu', extended_real=True)
assert integrate(meijerg([], [], [0], [], s*t)
* meijerg([], [], [mu/2], [-mu/2], t**2/4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x**s*meijerg([[], []], [[0], []], x),
(x, 0, oo), meijerg=False),
Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3*exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu)/(2*sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi)*sigma*(erf(mu/(2*sigma)) + 1)
assert c
i, _ = meijerint_definite(exp(-mu*x)*exp(sigma*x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1/(mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2*x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x - mu)/sigma)**2/2)/sqrt(2*pi*sigma**2),
x, -oo, oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x)*sin(x), x, 0, oo) == (Rational(1, 2), True)
# Test a bug
def res(n):
return (1/(1 + x**2)).diff(x, n).subs({x: 1})*(-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x)*sin(x + a), (x, 0, oo), meijerg=True)
) == sqrt(2)*sin(a + pi/4)/2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
* meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
* gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, Rational(1, 2))/4).expand()
# Test hyperexpand bug.
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + Rational(1, 2),
alpha/2 + 1)), ((0, 0, Rational(1, 2)), (-Rational(1, 2),)), alpha**2/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - Rational(1, 2))*((-1)**s + 1)*gamma(s/2 + Rational(1, 2))/2
def test_issue_4487():
from diofant import lowergamma, simplify
assert simplify(integrate(exp(-x) * x**y, x)) == lowergamma(y + 1, x)
def test_lowergamma():
from diofant import meijerg, exp_polar, I, expint
assert lowergamma(x, y).diff(y) == y**(x - 1) * exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert td(lowergamma(x, randcplx()), x)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
- meijerg([], [1, 1], [0, 0, x], [], y)
assert lowergamma(S.Half, x) == sqrt(pi) * erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False),
lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False),
lowergamma(S.Half - 3, x), x)
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(Rational(1, 3), lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4 * pi * I) * x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x),
conjugate(y))
assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))
assert lowergamma(
x, y).rewrite(expint) == -y**x * expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(
k, y).rewrite(expint) == -y**k * expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)